Analytical gradient calculation

An analytical gradient calculation is invariably faster than a numerical one. To repeat the argument from Chapter 14, with real wavefunction Hamiltonian H (including the field terms) and parameter a (where a is a component of the external electric field)... 

Analytical gradient calculations are quite effective when compared to the finite difference calculations. A factor of 20 applies in the present PCM formulation (Cossi et al., 1995). Although effective, this increment of efficiency is smaller, by a factor 10 or more, than the analogous speed up found for the calculations of gradients in vacuo. 

In general, analytic gradient calculations scale only weakly with the number of degrees of freedom and typically require roughly 1-3 times the CPU time as the energy calculation itself. Gradients for SCF and correlated methods may be evaluated from the formula... 

The Complete Active Space SCF (CASSCF) method [9] was used in most of the calculations. For Coov symmetry, the H surface was obtained considering as active the orbitals 2<7, 3(7-, 4(t, Itt, and 5o-, while the active electrons in six active orbitals. Analytical gradient calculations using the CASSCF wave-function were performed in order to evaluate the saddle point geometry and the minimum energy path. 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

Importantly for direct dynamics calculations, analytic gradients for MCSCF methods [124-126] are available in many standard quantum chemistiy packages. This is a big advantage as numerical gradients require many evaluations of the wave function. The evaluation of the non-Hellmann-Feynman forces is the major effort, and requires the solution of what are termed the coupled-perturbed MCSCF (CP-MCSCF) equations. The large memory requirements of these equations can be bypassed if a direct method is used [233]. Modem computer architectures and codes then make the evaluation of first and second derivatives relatively straightforward in this theoretical framework. 

Within some programs, the ROMPn methods do not support analytic gradients. Thus, the fastest way to run the calculation is as a single point energy calculation with a geometry from another method. If a geometry optimization must be done at this level of theory, a non-gradient-based method such as the Fletcher-Powell optimization should be used. 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

To show the principles involved in finding an analytical gradient expression consider an HF-LCAO calculation where the electronic energy comes to... 

Many ab initio packages use the two key equations given above in order to calculate the polarizabilities and hyperpolarizabilities. If analytical gradients are available, as they are for many levels of theory, then the quantities are calculated from the first or second derivative (with respect to the electric field), as appropriate. If analytical formulae do not exist, then numerical methods are used. 

Smooth COSMO solvation model. We have recently extended our smooth COSMO solvation model with analytical gradients [71] to work with semiempirical QM and QM/MM methods within the CHARMM and MNDO programs [72, 73], The method is a considerably more stable implementation of the conventional COSMO method for geometry optimizations, transition state searches and potential energy surfaces [72], The method was applied to study dissociative phosphoryl transfer reactions [40], and native and thio-substituted transphosphorylation reactions [73] and compared with density-functional and hybrid QM/MM calculation results. The smooth COSMO method can be formulated as a linear-scaling Green s function approach [72] and was applied to ascertain the contribution of phosphate-phosphate repulsions in linear and bent-form DNA models based on the crystallographic structure of a full turn of DNA in a nucleosome core particle [74],... 

As regards SCF and SCF-MI calculations, the GAMESS-US program was employed, in which the SCF-MI algorithm including evaluation of analytic gradient, geometry optimisation and force constant matrices computation is available [18,41,42]. 

However, the calculation requires different algorithms for each specific case. Hence, an explicit mathematical solution providing basing retention data (elution volumes or times, bandwidths, and resolution) is much more frequently used in analytical gradient chromatography, even at a cost of some simplifications [9-11]. 

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